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LCM of 24 and 32

LCM of 24 and 32 is the smallest number among all common multiples of 24 and 32. The first few multiples of 24 and 32 are (24, 48, 72, 96, 120, 144, 168, . . . ) and (32, 64, 96, 128, 160, . . . ) respectively. There are 3 commonly used methods to find LCM of 24 and 32 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 24 and 32
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 24 and 32?

Answer: LCM of 24 and 32 is 96.

Explanation:

The LCM of two non-zero integers, x(24) and y(32), is the smallest positive integer m(96) that is divisible by both x(24) and y(32) without any remainder.

Methods to Find LCM of 24 and 32

The methods to find the LCM of 24 and 32 are explained below.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 24 and 32 by Prime Factorization

Prime factorization of 24 and 32 is (2 × 2 × 2 × 3) = 2³ × 3¹ and (2 × 2 × 2 × 2 × 2) = 2⁵ respectively. LCM of 24 and 32 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁵ × 3¹ = 96.
Hence, the LCM of 24 and 32 by prime factorization is 96.

LCM of 24 and 32 by Division Method

To calculate the LCM of 24 and 32 by the division method, we will divide the numbers(24, 32) by their prime factors (preferably common). The product of these divisors gives the LCM of 24 and 32.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 24 and 32. Write this prime number(2) on the left of the given numbers(24 and 32), separated as per the ladder arrangement.
Step 2: If any of the given numbers (24, 32) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 24 and 32 is the product of all prime numbers on the left, i.e. LCM(24, 32) by division method = 2 × 2 × 2 × 2 × 2 × 3 = 96.

LCM of 24 and 32 by Listing Multiples

To calculate the LCM of 24 and 32 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 24 (24, 48, 72, 96, 120, 144, 168, . . . ) and 32 (32, 64, 96, 128, 160, . . . . )
Step 2: The common multiples from the multiples of 24 and 32 are 96, 192, . . .
Step 3: The smallest common multiple of 24 and 32 is 96.

∴ The least common multiple of 24 and 32 = 96.

☛ Also Check:

LCM of 24 and 32 Examples

Example 1: Verify the relationship between GCF and LCM of 24 and 32.

Solution:

The relation between GCF and LCM of 24 and 32 is given as,
LCM(24, 32) × GCF(24, 32) = Product of 24, 32
Prime factorization of 24 and 32 is given as, 24 = (2 × 2 × 2 × 3) = 2³ × 3¹ and 32 = (2 × 2 × 2 × 2 × 2) = 2⁵
LCM(24, 32) = 96
GCF(24, 32) = 8
LHS = LCM(24, 32) × GCF(24, 32) = 96 × 8 = 768
RHS = Product of 24, 32 = 24 × 32 = 768
⇒ LHS = RHS = 768
Hence, verified.
Example 2: The product of two numbers is 768. If their GCD is 8, what is their LCM?

Solution:

Given: GCD = 8
product of numbers = 768
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 768/8
Therefore, the LCM is 96.
The probable combination for the given case is LCM(24, 32) = 96.
Example 3: The GCD and LCM of two numbers are 8 and 96 respectively. If one number is 32, find the other number.

Solution:

Let the other number be b.
∵ GCD × LCM = 32 × b
⇒ b = (GCD × LCM)/32
⇒ b = (8 × 96)/32
⇒ b = 24
Therefore, the other number is 24.

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FAQs on LCM of 24 and 32

What is the LCM of 24 and 32?

The LCM of 24 and 32 is 96. To find the LCM (least common multiple) of 24 and 32, we need to find the multiples of 24 and 32 (multiples of 24 = 24, 48, 72, 96; multiples of 32 = 32, 64, 96, 128) and choose the smallest multiple that is exactly divisible by 24 and 32, i.e., 96.

What is the Least Perfect Square Divisible by 24 and 32?

The least number divisible by 24 and 32 = LCM(24, 32)
LCM of 24 and 32 = 2 × 2 × 2 × 2 × 2 × 3 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 24 and 32 = LCM(24, 32) × 2 × 3 = 576 [Square root of 576 = √576 = ±24]
Therefore, 576 is the required number.

What is the Relation Between GCF and LCM of 24, 32?

The following equation can be used to express the relation between GCF and LCM of 24 and 32, i.e. GCF × LCM = 24 × 32.

If the LCM of 32 and 24 is 96, Find its GCF.

LCM(32, 24) × GCF(32, 24) = 32 × 24
Since the LCM of 32 and 24 = 96
⇒ 96 × GCF(32, 24) = 768
Therefore, the greatest common factor (GCF) = 768/96 = 8.

How to Find the LCM of 24 and 32 by Prime Factorization?

To find the LCM of 24 and 32 using prime factorization, we will find the prime factors, (24 = 2 × 2 × 2 × 3) and (32 = 2 × 2 × 2 × 2 × 2). LCM of 24 and 32 is the product of prime factors raised to their respective highest exponent among the numbers 24 and 32.
⇒ LCM of 24, 32 = 2⁵ × 3¹ = 96.

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